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I'm trying to understand how a prediction from a multinomial logit model compares to a prediction from many logit models. A few videos and wikipedia have lead to me to believe that the categorical prediction from a multinomial model is equivalent to the prediction of the logit model for which the prediction is the highest.

As an example, I have 4 specific (ranked) outcomes: 1,2,3,4. Each item in the dataset has continuous independent variables and 4 different indicator variables to indicate if the response is 1,2,3, or 4. I then run a logistic regression for each response and obtain 4 sets of fitted betas. Each of these sets of coefficients are applied to a new X that I want to predict. I can then make four predictions of the dependant variable y, one for each outcome. Given the link function, this will be transformed into a probability of how likely this new X is to fall into each of the 4 outcomes.

If, for example, the single logit model predicts that outcome 3 will occur with a 29% probability, and that this is the higher than outcomes 1,2, or 4, is this the prediction that the multinomial logit model would provide as well?

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There is an approximate relationship between logistic regressions and a multinomial logitsitc regression, but not the one you describe, see for example: Several logistic regressions vs multinomial regression

I suspect that the main problem you have is the concept that multinomial logits give you predicted probabilities on all outcomes rather than a predicted $y$ as in linear regression. You of course can apply some arbitrary cut-off to say it predicts outcome 3, but that usually works extremely poorly. For example, if one category is large, then you always predict that large category, and the explanatory variables add nothing at all.

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